UN CO RR EC TE D PR O O F 1 ARAS 2 Preconditioning Technique for CFD Industrial 2 Cases 3
نویسنده
چکیده
The convergence rate of a Krylov method such as the Generalized Conjugate Resid10 ual (GCR) [6] method, to solve a linear system Au = f , A = (ai j) ∈ Rm×m,u ∈ 11 R m, f ∈ Rm, decreases with increasing condition number κ2(A) = ||A||2||A−1||2 12 of the non singular matrix A. Left preconditioning techniques consist of solving 13 M−1Au=M−1 f such that κ2(M−1A)<<κ2(A). The Additive Schwarz (AS) precon14 ditioning is built from the adjacency graph G =(W,E) of A, where W = {1,2, . . . ,m} 15 and E = {(i, j) : ai j = 0} are the edges and vertices of G. Starting with a non16 overlapping partition W =∪ i=1Wi,0 and δ ≥ 0 given, the overlapping partition {Wi,δ} 17 is obtained defining p partitions Wi,δ ⊃Wi,δ−1 by including all the immediate neigh18 boring vertices of the vertices in the partition Wi,δ−1. Then the restriction opera19 tor Ri,δ from W to Wi,δ defines the local operator Ai,δ = Ri,δAR T i,δ ,Ai,δ ∈ Rmi,δ×mi,δ 20 on Wi,δ . The AS preconditioning writes: M −1 AS,δ = p ∑ i=1 Ri,δA −1 i,δ Ri,δ . Introducing R̃i,δ 21 the restriction matrix on a non-overlapping subdomain Wi,0, the Restricted Additive 22 Schwarz (RAS) iterative process [2] writes: 23
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1 Sandia National Labs, Albuquerque, NM 87185 ([email protected]). Sandia National 5 Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, 6 a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of 7 Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. 8 2 The University of Texas at Austin, Institute fo...
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1 Department of Mathematics and Center for Computation & Technology, Louisiana State 5 University, Baton Rouge, LA 70803, USA. [email protected] 6 2 Department of Mathematics and Center for Computation & Technology, Louisiana State 7 University, Baton Rouge, LA 70803, USA. [email protected] 8 3 Department of Mathematics and Center for Computation & Technology, Louisiana State 9 University, B...
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